10 MAXIMUM AND MINIMUM
|
f(x) | = | x2 − 6x + 5 |
f(3) | = | 32 − 6· 3 + 5 |
= | −4. |
The extreme value is −4. To see whether it is a maximum or a minimum, in this case we can simply look at the graph.
f(x) is a parabola, and we can see that the turning point is a minimum.
By finding the value of x where the derivative is 0, then, we have discovered that the vertex of the parabola is at (3, −4).
But we will not always be able to look at the graph. The algebraic condition for a minimum is that f '(x) changes sign from − to + . We see this at the points E, B, F above. The value of the slope is increasing.
Now to say that the slope is increasing, is to say that, at a critical value, the second derivative (Lesson 9) -- which is rate of change of the slope -- is positive.
Again, here is f(x):
f(x) | = | x2 − 6x + 5. |
f '(x) | = | 2x − 6. |
f ''(x) | = | 2. |
f '' evaluated at the critical value 3 -- f''(3) = 2 -- is positive. This tells us algebraically that the critical value 3 determines a minimum.
Sufficient conditions
We can now state these sufficient conditions for extreme values of a function at a critical value a:
The function has a minimum value at x = a if f '(a) = 0
and f ''(a) = a positive number.
The function has a maximum value at x = a if f '(a) = 0
and f ''(a) = a negative number.
In the case of the maximum, the slope of the tangent is decreasing -- it is going from positive to negative. We can see that at the points C, A, D.
Example 2. Let f(x) = 2x3− 9x2 + 12x − 3.
Are there any extreme values? First, are there any critical values -- solutions to f '(x) = 0 -- and do they determine a maximum or a minimum? And what are the coördinates on the graph of that maximum or minimum? Where are the turning points?
Solution. f '(x) = 6x2 − 18x + 12 | = | 6(x2 − 3x + 2) |
= | 6(x − 1)(x − 2) | |
= | 0 |
implies:
x = 1 or x = 2.
Those are the critical values. Does each one determine a maximum or does it determine a minimum? To answer, we must evaluate the second derivative at each value.
f '(x) | = | 6x2 − 18x + 12. |
f ''(x) | = | 12x − 18. |
f ''(1) | = | 12 − 18 = −6. |
The second derivative is negative. The function therefore has a maximum at x = 1.
To find the y-coördinate -- the extreme value -- at that maximum we evaluate f(1):
f(x) | = | 2x3− 9x2 + 12x − 3 |
f(1) | = | 2 − 9 + 12 − 3 |
= | 2. |
The maximum occurs at the point (1, 2).
Next, does x = 2 determine a maximum or a minimum?
f ''(x) | = | 12x − 18. |
f ''(2) | = | 24 − 18 = 6. |
The second derivative is positive. The function therefore has a minimum at x = 2.
To find the y-coördinate -- the extreme value -- at that minimum, we evaluate f(2):
f(x) | = | 2x3 − 9x2 + 12x − 3. |
f(2) | = | 16 − 36 + 24 − 3 |
= | 1. |
The minimum occurs at the point (2, 1).
Here in fact is the graph of f(x):
Solutions to f ''(x) = 0 indicate a point of inflection at those solutions, not a maximum or minimum. An example is y = x3. y'' = 6x = 0 implies x = 0. But x = 0 is a point of inflection in the graph of y = x3, not a maximum or minimum.
Another example is y = sin x. The solutions to y'' = 0 are the multiplies of π, which are points of inflection.
Problem 1. Find the coördinates of the vertex of the parabola,
y = x2 − 8x + 1.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
y' = 2x − 8 = 0.
That implies x = 4. That's the x-coördinate of the vertex. To find the y-coördinate, evaluate y at x = 4:
y = 42 − 8· 4 + 1 = −15.
The vertex is at (4, −15).
Problem 2. Examine each function for maxima and minima.
a) y = x3 − 3x2 + 2.
y' = 3x2 − 6x = 3x(x − 2) = 0 implies
x = 0 or x = 2.
y''(x) = 6x − 6.
y''(0) = −6.
The second derivative is negative. That means there is a maximum at x = 0. That maximum value is
y(0) = 2.
Next,
y''(2) = 12 − 6 = 6.
The second derivative is positive. That means there is a minimum at x = 2. That minimum value is
y(2) = 23 − 3· 22 + 2 = 8 − 12 + 2 = −2.
b) y = −2x3 − 3x2 + 12 x + 10.
At x = 1 there is a maximum of y = 17.
At x = −2 there is a minimum of y = −10.
c) y = 2x3 + 3x2 + 12 x − 4.
Since f '(x) = 0 has no real solutions, there are no extreme values.
d) y = 3x4− 4x3 − 12x2 + 2.
At x = 0 there is a maximum of y = 2.
At x = −1 there is a minimum of y = −3.
At x = 2 there is a minimum of y = −30.
Next Lesson: Applications of maximum and minimum values
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2021 Lawrence Spector
Questions or comments?
E-mail: teacher@themathpage.com